Modifications for the Differential Evolution Algorithm
Abstract
:1. Introduction
2. Modifications
2.1. The Base Algorithm
Algorithm 1: DE algorithm. 

2.2. The New Termination Rule
2.3. The New Differential Weight
3. Experiments
3.1. Test Functions
 Bf1 (Bohachevsky 1) function defined as:$$f\left(x\right)={x}_{1}^{2}+2{x}_{2}^{2}\frac{3}{10}cos\left(3\pi {x}_{1}\right)\frac{4}{10}cos\left(4\pi {x}_{2}\right)+\frac{7}{10}$$
 Bf2 (Bohachevsky 2) function defined as:$$f\left(x\right)={x}_{1}^{2}+2{x}_{2}^{2}\frac{3}{10}cos\left(3\pi {x}_{1}\right)cos\left(4\pi {x}_{2}\right)+\frac{3}{10}$$
 Branin function. The function is defined by $f\left(x\right)={\left({x}_{2}\frac{5.1}{4{\pi}^{2}}{x}_{1}^{2}+\frac{5}{\pi}{x}_{1}6\right)}^{2}+10\left(1\frac{1}{8\pi}\right)cos\left({x}_{1}\right)+10$ with $5\le {x}_{1}\le 10,\phantom{\rule{4pt}{0ex}}0\le {x}_{2}\le 15$. The value of global minimum is 0.397887.with $x\in {[10,10]}^{2}$. The value of global minimum is −0.352386.
 CM function. The Cosine Mixture function is given by the equation:$$f\left(x\right)=\sum _{i=1}^{n}{x}_{i}^{2}\frac{1}{10}\sum _{i=1}^{n}cos\left(5\pi {x}_{i}\right)$$
 Camel function. The function is given by:$$f\left(x\right)=4{x}_{1}^{2}2.1{x}_{1}^{4}+\frac{1}{3}{x}_{1}^{6}+{x}_{1}{x}_{2}4{x}_{2}^{2}+4{x}_{2}^{4},\phantom{\rule{1.em}{0ex}}x\in {[5,5]}^{2}$$
 Easom function. The function is given by the equation:$$f\left(x\right)=cos\left({x}_{1}\right)cos\left({x}_{2}\right)exp\left({\left({x}_{2}\pi \right)}^{2}{\left({x}_{1}\pi \right)}^{2}\right)$$
 Exponential function, defined as:$$f\left(x\right)=exp\left(0.5\sum _{i=1}^{n}{x}_{i}^{2}\right),\phantom{\rule{1.em}{0ex}}1\le {x}_{i}\le 1$$The global minimum is located at ${x}^{*}=(0,0,...,0)$ with value $1$. In our experiments we used this function with $n=2,4,8,16,32$.
 Goldstein and Price functionThe function is given by the equation:$$\begin{array}{ccc}\hfill f\left(x\right)& =& \left[1+{\left({x}_{1}+{x}_{2}+1\right)}^{2}\right.\hfill \\ & & \left(1914{x}_{1}+3{x}_{1}^{2}14{x}_{2}+6{x}_{1}{x}_{2}+3{x}_{2}^{2}\right)]\times \hfill \\ & & [30+{\left(2{x}_{1}3{x}_{2}\right)}^{2}\hfill \\ & & \left(1832{x}_{1}+12{x}_{1}^{2}+48{x}_{2}36{x}_{1}{x}_{2}+27{x}_{2}^{2}\right)]\hfill \end{array}$$
 Griewank2 function. The function is given by:$$f\left(x\right)=1+\frac{1}{200}\sum _{i=1}^{2}{x}_{i}^{2}\prod _{i=1}^{2}\frac{cos\left({x}_{i}\right)}{\sqrt{\left(i\right)}},\phantom{\rule{1.em}{0ex}}x\in {[100,100]}^{2}$$The global minimum is located at the ${x}^{*}=(0,0,...,0)$ with value 0.
 Gkls function. $f\left(x\right)=\mathrm{Gkls}(x,n,w)$, is a function with w local minima, described in [71] with $x\in {[1,1]}^{n}$ and n a positive integer between 2 and 100. The value of the global minimum is −1 and in our experiments we have used $n=2,3$ and $w=50,\phantom{\rule{4pt}{0ex}}100$.
 Hansen function. $f\left(x\right)={\sum}_{i=1}^{5}icos\left[(i1){x}_{1}+i\right]{\sum}_{j=1}^{5}jcos\left[(j+1){x}_{2}+j\right]$, $x\in $${[10,10]}^{2}$.
 Hartman 3 function. The function is given by:$$f\left(x\right)=\sum _{i=1}^{4}{c}_{i}exp\left(\sum _{j=1}^{3}{a}_{ij}{\left({x}_{j}{p}_{ij}\right)}^{2}\right)$$$$p=\left(\begin{array}{ccc}0.3689& 0.117& 0.2673\\ 0.4699& 0.4387& 0.747\\ 0.1091& 0.8732& 0.5547\\ 0.03815& 0.5743& 0.8828\end{array}\right)$$
 Hartman 6 function.$$f\left(x\right)=\sum _{i=1}^{4}{c}_{i}exp\left(\sum _{j=1}^{6}{a}_{ij}{\left({x}_{j}{p}_{ij}\right)}^{2}\right)$$$$p=\left(\begin{array}{cccccc}0.1312& 0.1696& 0.5569& 0.0124& 0.8283& 0.5886\\ 0.2329& 0.4135& 0.8307& 0.3736& 0.1004& 0.9991\\ 0.2348& 0.1451& 0.3522& 0.2883& 0.3047& 0.6650\\ 0.4047& 0.8828& 0.8732& 0.5743& 0.1091& 0.0381\end{array}\right)$$
 Potential function. The molecular conformation corresponding to the global minimum of the energy of N atoms interacting via the LennardJones potential [72] is used as a test case here. The function to be minimized is given by:$${V}_{LJ}\left(r\right)=4\u03f5\left[{\left(\frac{\sigma}{r}\right)}^{12}{\left(\frac{\sigma}{r}\right)}^{6}\right]$$In the current experiments three different cases were studied: $N=3,\phantom{\rule{4pt}{0ex}}4,\phantom{\rule{4pt}{0ex}}5.$
 Rastrigin function. The function is given by:$$f\left(x\right)={x}_{1}^{2}+{x}_{2}^{2}cos\left(18{x}_{1}\right)cos\left(18{x}_{2}\right),\phantom{\rule{1.em}{0ex}}x\in {[1,1]}^{2}$$The global minimum is located at ${x}^{*}=(0,0)$ with value −2.0.
 Rosenbrock function.This function is given by:$$f\left(x\right)=\sum _{i=1}^{n1}\left(100{\left({x}_{i+1}{x}_{i}^{2}\right)}^{2}+{\left({x}_{i}1\right)}^{2}\right),\phantom{\rule{1.em}{0ex}}30\le {x}_{i}\le 30.$$The global minimum is located at the ${x}^{*}=(0,0,...,0)$ with $f\left({x}^{*}\right)=0$. In our experiments we used this function with $n=4,\phantom{\rule{4pt}{0ex}}8,\phantom{\rule{4pt}{0ex}}16$.
 Shekel 7 function.
 Shekel 5 function.
 Shekel 10 function.
 Sinusoidal function. The function is given by:$$f\left(x\right)=\left(2.5\prod _{i=1}^{n}sin\left({x}_{i}z\right)+\prod _{i=1}^{n}sin\left(5\left({x}_{i}z\right)\right)\right),\phantom{\rule{1.em}{0ex}}0\le {x}_{i}\le \pi .$$The global minimum is located at ${x}^{*}=(2.09435,2.09435,...,2.09435)$ with $f\left({x}^{*}\right)=3.5$. In our experiments we used $n=4,8,16,32$ and $z=\frac{\pi}{6}$ and the corresponding functions are denoted by the labels SINU4, SINU8, SINU16 and SINU32, respectively.
 Test2N function. This function is given by the equation:$$f\left(x\right)=\frac{1}{2}\sum _{i=1}^{n}{x}_{i}^{4}16{x}_{i}^{2}+5{x}_{i},\phantom{\rule{1.em}{0ex}}{x}_{i}\in [5,5].$$The function has ${2}^{n}$ in the specified range and in our experiments we used $n=4,5,6,7$. The corresponding values of global minimum is −156.664663 for $n=4$, −195.830829 for $n=5$, −234.996994 for $n=6$ and −274.163160 for $n=7$.
 Test30N function. This function is given by:$$f\left(x\right)=\frac{1}{10}{sin}^{2}\left(3\pi {x}_{1}\right)\sum _{i=2}^{n1}\left({\left({x}_{i}1\right)}^{2}\left(1+{sin}^{2}\left(3\pi {x}_{i+1}\right)\right)\right)+{\left({x}_{n}1\right)}^{2}\left(1+{sin}^{2}\left(2\pi {x}_{n}\right)\right)$$
3.2. Experimental Results
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Parameter  Value 

NP  10n 
F  0.8 
CR  0.9 
M  20 
$\u03f5$  ${10}^{4}$ 
Function  Static  Ali  Proposed 

BF1  1142  1431  847 
BF2  1164  1379  896 
BRANIN  984  816  707 
CM4  3590  7572  2079 
CAMEL  1094  18,849  685 
EASOM  1707  2014  1327 
EXP2  532  323  449 
EXP4  2421  1019  1494 
EXP8  15,750  3670  5632 
EXP16  160,031  15,150  21,416 
EXP32  320,039  152,548  77,936 
GKLS250  784  944  614 
GKLS2100  772  1531  599 (0.97) 
GKLS350  1906 (0.93)  3263  1275 (0.93) 
GKLS3100  1883  3539  1373 
GOLDSTEIN  988  818  769 
GRIEWANK2  1299 (0.97)  1403  883 (0.93) 
HANSEN  2398  2968  1400 
HARTMAN3  1448  836  1050 
HARTMAN6  9489(0.97)  4015(0.97)  4667(0.80) 
POTENTIAL3  90,027  89,776  21,824 
POTENTIAL4  120,387 (0.97)  120,405 (0.33)  45,705 (0.97) 
POTENTIAL5  150,073  150,104  83,342 
RASTRIGIN  1246  1098 (0.93)  871 
ROSENBROCK4  6564  9695  4499 
ROSENBROCK8  44,240  72,228  13,959 
ROSENBCROK16  160,349 (0.90)  160,538 (0.60)  53,594 
SHEKEL5  5524  3810  3057 (0.83) 
SHEKEL7  5266  3558  2992 (0.87) 
SHEKEL10  5319  3379  3076 
TEST2N4  4200  1980  2592 
TEST2N5  7357  2957  4055 
TEST2N6  12,074  4159  5836 
TEST2N7  18,872  5490  7904 
SINU4  3270  1855  2216 
SINU8  23,108  6995  8135 
SINU16  160,092  36,044  30,943 
SINU32  213,757 (0.70)  160,536 (0.53)  83,369 (0.80) 
TEST30N3  1452  1732  959 
TEST30N4  1917  2287  1378 
Total  1,564,515 (0.97)  1,062,714 (0.96)  506,404 (0.98) 
Function  Static  Ali  Proposed 

BF1  996  1124  889 
BF2  926  1026  816 
BRANIN  878  900  730 
CM4  1148 (0.70)  1991  1103 
CAMEL  1049  904 (0.93)  846 
EASOM  447  448  446 
EXP2  470  461  467 
EXP4  915  903  892 
EXP8  1797  3558  1796 
EXP16  3578  7082  3521 
EXP32  7082  14,125  7022 
GKLS250  498  576  493 
GKLS2100  533  884 (0.97)  515 
GKLS350  823  1130 (0.93)  814 (0.97) 
GKLS3100  858  1495 (0.97)  829 (0.93) 
GOLDSTEIN  945  993  915 
GRIEWANK2  947  921  826 
HANSEN  2104  1949  1479 
HARTMAN3  1017  1005  952 
HARTMAN6  4679 (0.90)  3744 (0.97)  3128 (0.87) 
POTENTIAL3  21,473  2284  8197 
POTENTIAL4  44,191 (0.43)  3098 (0.33)  24,659 (0.97) 
POTENTIAL5  75,910  3443  52,664 
RASTRIGIN  841  994  777 
ROSENBROCK4  4934  7192  3300 
ROSENBROCK8  29,583  49,696  10,907 
ROSENBCROK16  160,349  160,538 (0.60)  38,315 
SHEKEL5  4389 (0.97)  4266  2839 (0.83) 
SHEKEL7  3905  3685  2668 
SHEKEL10  4049  3548  2629 
TEST2N4  2785  2275  2221 
TEST2N5  4481  3170  3122 
TEST2N6  6852  4286  4296 
TEST2N7  11971  5701  6267 
SINU4  2322  1987  1755 
SINU8  9990  6156  5113 
SINU16  6892  3628 (0.97)  16,905 
SINU32  7235 (0.80)  7438 (0.83)  7218 
TEST30N3  1033  1098  951 
TEST30N4  1355  1444  1285 
Total  432,610 (0.98)  321,166 (0.96)  224,567 (0.99) 
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Charilogis, V.; Tsoulos, I.G.; Tzallas, A.; Karvounis, E. Modifications for the Differential Evolution Algorithm. Symmetry 2022, 14, 447. https://doi.org/10.3390/sym14030447
Charilogis V, Tsoulos IG, Tzallas A, Karvounis E. Modifications for the Differential Evolution Algorithm. Symmetry. 2022; 14(3):447. https://doi.org/10.3390/sym14030447
Chicago/Turabian StyleCharilogis, Vasileios, Ioannis G. Tsoulos, Alexandros Tzallas, and Evangelos Karvounis. 2022. "Modifications for the Differential Evolution Algorithm" Symmetry 14, no. 3: 447. https://doi.org/10.3390/sym14030447